Our edge index is defined as a spectral flow, a powerful concept that has been used for bulk-edge correspondences in 2D discrete models of topological insulators in [1]}, [2]}, [3]}, and 3D Weyl semimetals in [4]}, [5]}. Along the above-mentioned loop, it counts the signed number of crossings of edge mode branches with a fiducial line. Notice that such a quantity was ill-defined in [6]} due to the non-compactness of the longitudinal momentum parameter. We then provide a more general definition of spectral flow due to Phillips [7]}, [8]}, and show that our edge index is stable against relatively compact perturbations. Finally, we explore the whole spectral flow structure of the model along any loop of the parameter space, and show that they are all related to the Chern number. Notice that spectral flow
